ECE380 Digital Logic

Transcription

1 ECE38 Digital Logic State Minimization Dr. D. J. Jackson Lecture 32- State minimization For simple FSMs, it is easy to see from the state diagram that the number of states used is the minimum possible FSMs for counters are good examples For more complex FSMs, it is likely that an initial state diagram may have more states than are necessary to perform a required function Minimizing states is of interest because fewer states implies fewer flip-flops to implement the circuit Complexity of combinational logic may also be reduced Instead of trying to show which states are equivalent, it is often easier to show which states are definitely not equivalent This can be exploited to define a minimization procedure Dr. D. J. Jackson Lecture 32-2

2 State equivalence Definition: Two states S i and S j are equivalent if and only if for every possible input sequence, the same output sequence will be produced regardless of whether S i or S j is the initial state If an input w= is applied to an FSM in state S i and the FSM transitions to state S u, then S u is termed a -successor of S i Similarly, if w= and the FSM transitions to S y, then S y is a -successor of S i The successors of S i are its k-successors With one input, k can only be or, but if there are multiple inputs then k represents all the valuations of the inputs Dr. D. J. Jackson Lecture 32-3 Partitioning minimization From the equivalence definition, if S i and S j are equivalent then their corresponding k-successors are also equivalent Using this, we can construct a minimization procedure that involves considering the states of the machine as a set and then breaking that set into partitions that comprise subsets that are definitely not equivalent Definition: A partition consists of one or more blocks, where each block comprises a subset of states that may be equivalent, but the states in one block are definitely not equivalent to the states in other blocks Dr. D. J. Jackson Lecture

3 Partition minimization example Consider the following state table Present Next state Output state w = w = z A B C B D F C F E D B G E F C F E D G F G An initial partition contains all the states in a single block P =(ABCDEFG) Dr. D. J. Jackson Lecture 32-5 Partition minimization example The next partition separates the states that have different outputs P 2 =(ABD)(CEFG) Now consider all - and -successors of the states in each block For (ABD), the -successors are (BDB) Since these are all in the same block we must still consider A, B, and D equivalent The -successors of (ABD) are (CFG) We must still consider A, B, and D equivalent Now consider the (CEFG) block Dr. D. J. Jackson Lecture

4 Partition minimization example P 2 =(ABD)(CEFG) For (CEFG), the -successors are (FEFF) which are all in the same block in P 2 C, E, F, and G must still be considered equivalent The -successors of (CEFG) are (ECDG) Since these are not in the same block in P 2 then at least one of the states in (CEFG) is not equivalent to the others State F must be different from C, E and G because its - successor,d, is in a different block than E, C, and G Therefore, P 3 =(ABD)(CEG)(F) At this point, we know that state F is unique since it is in a block by itself Dr. D. J. Jackson Lecture 32-7 Partition minimization example P 3 =(ABD)(CEG)(F) The process repeats yielding the following -successors of (ABD) are (BDB) A, B and D are still considered equivalent -successors of (ABD) are (CFG), which are not in the same block B cannot be equivalent to A and D since F is in a different block than C and G The - and -successors of (CEG) are (FFF) and (ECG) C, E, and G must still be considered equivalent Thus, P 4 =(AD)(B)(CEG)(F) Dr. D. J. Jackson Lecture

5 Partition minimization example If we repeat the process to check the - and -successors of the blocks (AD) and (CEG), we find that P 5 =(AD)(B)(CEG)(F) Since P 5 =P 4 and no new blocks are generated, it follows that states in each block are equivalent A and D are equivalent C, E, and G are equivalent Dr. D. J. Jackson Lecture 32-9 Partition minimization example The state table can be rewritten, removing the rows for D, E and G and replacing all occurrences of D with A and all occurrences of E or G with C The resulting state table is as follows Present Nextstate Output state w = w = z A B C B A F C F C F C A Dr. D. J. Jackson Lecture 32-5

6 Design Example Determine which states (if any) are equivalent in the following Moore state diagram reset A/ B/ C/ D/ E/ F/ Dr. D. J. Jackson Lecture 32-6